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Inverse electromagnetic scattering problems by a doubly periodic structure

Consider the problem of scattering of electromagnetic waves by a doubly periodic structure. The medium above the structure is assumed to be inhomogeneous characterized completely by an index of refraction. Below the structure is a perfect conductor or an imperfect conductor partially coated with a dielectric. Having established the well-posedness of the direct problem by the variational approach, we prove the uniqueness of the inverse problem, that is, the unique determination of the doubly periodic grating with its physical property and the index of refraction from a knowledge of the scattered near field by a countably infinite number of incident quasi-periodic electromagnetic waves. A key ingredient in our proofs is a novel mixed reciprocity relation derived in this paper.